1 . 平面直角坐标系
中,已知椭圆
:
的离心率为
,且点(
,
)在椭圆
上.
(Ⅰ)求椭圆
的方程;
(Ⅱ)设椭圆
:
,
为椭圆
上任意一点,过点
的直线
交椭圆
于
两点,射线
交椭圆
于点
.
(ⅰ)求
的值;
(ⅱ)求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/189304c2e50f0288073adb24314a3708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(Ⅰ)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(Ⅱ)设椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bccf89a4edf790cc2e1418d1eb1e994.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15b256345d7109e081b7c895591e995d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef49a3ca580a144cc65a609c167facc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(ⅰ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3b4394cae20185821fadfced2c5118b.png)
(ⅱ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c60b5065275b6bf1e5597d2f544e45a.png)
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7卷引用:2015-2016学年湖南株洲二中高二上第三次月考文数学卷
14-15高二上·湖南益阳·阶段练习
2 . 设
、
是椭圆
上的两点,已知向量
,
,若
且椭圆的离心率
,短轴长为
,
为坐标原点.
(1)求椭圆的方程;
(2)若直线
过椭圆的焦点
(
为半焦距),求直线
的斜率
的值;
(3)试问:
的面积是否为定值?如果是,请给予证明;如果不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c9bebea391a1f9956dfcca98d9d1f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d02faad03bf6e39163e973121db5443.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03857683a18d59f196ae443421db2904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/169488d6e49c88d14e9d23419c255568.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5de85df85401e7e8da683ea4a784963c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(1)求椭圆的方程;
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/925752b294d622b2386a885ff67382bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)试问:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
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6卷引用:2014-2015学年湖南省益阳市箴言中学高二9月月考理科数学试卷
3 . 如图,
为坐标原点,椭圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a259dddccc3cfe8f81735cd5834785d.png)
的左右焦点分别为
,离心率为
;双曲线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cf1bce17440e83c1b735d2954010a2a.png)
的左右焦点分别为
,离心率为
,已知
,且
.
(1)求
的方程;
(2)过
点作
的不垂直于
轴的弦
,
为
的中点,当直线
与
交于
两点时,求四边形
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a259dddccc3cfe8f81735cd5834785d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed6b9540857e386651e191a0a5b5a98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d33558881906c228c262ff8024dcfc4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cf1bce17440e83c1b735d2954010a2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/626706e779756baf8f7aa4cd276d2017.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33a99190a8fd29c36d5a002e3197cc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e749c051ebeba72d9873b4f31c8ba7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db4aed953f852a8f9eab33645b2078dc.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21e9feabc99f62ee569b460e61526e2e.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d4ab45e8e8f0084d8d90a4c1233d86.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/28/eb8915cc-1337-4d04-a350-edd99f29bb3a.png?resizew=262)
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6卷引用:2014年全国普通高等学校招生统一考试理科数学(湖南卷)
2014年全国普通高等学校招生统一考试理科数学(湖南卷)四川省成都外国语学校2017-2018学年高二下学期入学考试数学(理)试题(已下线)专题30 圆锥曲线三角形面积与四边形面积题型全归类-2(已下线)重难点突破17 圆锥曲线中参数范围与最值问题(八大题型)重庆市第十八中学2023-2024学年高二上学期12月学习能力摸底数学试题(已下线)专题24 解析几何解答题(理科)-3
2014·广东湛江·一模
名校
解题方法
4 . 如图,点
是椭圆
的一个顶点,
的长轴是圆
的直径,
、
是过点
且互相垂直的两条直线,其中
交圆
于
、
两点,
交椭圆
于另一点
.
![](https://img.xkw.com/dksih/QBM/2014/4/28/1571681726406656/1571681732419584/STEM/51d6c4e654fd4e8897694465f0e6c1cc.png?resizew=261)
(1)求椭圆
的方程;
(2)求
面积的最大值及取得最大值时直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6951479694aec937a712901634a5a4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/471257463df494d995b2cbb686e5b027.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a3ef6da64080039ca5541c7d6b23d49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://img.xkw.com/dksih/QBM/2014/4/28/1571681726406656/1571681732419584/STEM/51d6c4e654fd4e8897694465f0e6c1cc.png?resizew=261)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db5e1441a49e782ff0ef46e776cde06a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
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9卷引用:湖南省株洲市醴陵第二中学、醴陵第四中学2018届高三上学期两校期中联考数学(理)试题
湖南省株洲市醴陵第二中学、醴陵第四中学2018届高三上学期两校期中联考数学(理)试题(已下线)2014届广东省湛江市高三高考模拟测试二理科数学试卷(已下线)2014届广东省湛江市高三高考模拟测试二文科数学试卷四川省成都市双流中学2017高二上学期期中考试数学试题安徽省六安市舒城中学2017-2018学年高二下学期第一次统考(开学考试)数学(文)试题2020届山东省潍坊市高三下学期开学考试数学试题(已下线)强化卷03(3月)-冲刺2020高考数学之拿高分题目强化卷(山东专版)(已下线)备战2020年高考数学之考场再现(山东专版)05(已下线)第8篇——平面解析几何-新高考山东专题汇编
2013·广东惠州·一模
5 . 已知中心在原点O,焦点在x轴上,离心率为
的椭圆过点
.
(1)求椭圆的方程;
(2)设不过原点O的直线与该椭圆交于P、Q两点,满足直线OP,PQ,OQ的斜率依次成等比数列,求
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d59419fff2c12efda787ef2c195af18.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/8/0bf60613-2af7-4294-b30f-3855ccdeebee.png?resizew=205)
(1)求椭圆的方程;
(2)设不过原点O的直线与该椭圆交于P、Q两点,满足直线OP,PQ,OQ的斜率依次成等比数列,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1f0417d8269f01d8e0bc1a8756e2ac.png)
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6 . 已知圆M:(x+1)2+y2=1,圆N:(x-1)2+y2=9,动圆P与圆M外切并与圆N内切,圆心P的轨迹为曲线 C
(1)求C的方程;
(2)l是与圆P,圆M都相切的一条直线,l与曲线C交于A,B两点,当圆P的半径最长时,求|AB|.
(1)求C的方程;
(2)l是与圆P,圆M都相切的一条直线,l与曲线C交于A,B两点,当圆P的半径最长时,求|AB|.
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19卷引用:2020届湖南省长沙市长郡中学高三第二次月考数学(文)试题
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7 . 已知椭圆的长轴长为
,焦点是
,点
到直线
的距离为
,过点
且倾斜角为锐角的直线
与椭圆交于A、B两点,使得|
=3|![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7f2b5b1e9ef7dd60486b550eb4cbec1.png)
.
(1)求椭圆的标准方程;
(2)求直线l的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878e89b6eca35e34c863e832a2c661db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc1d75519e5696817df077e7c9b00a9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb54f2b3f5f909a41d8171fef91181c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827ccf0c04aa941ba20d5f4c6068b46b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f04dfa7d041711f992098cdd42efe241.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7f2b5b1e9ef7dd60486b550eb4cbec1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dc24eb7a51f79bd4bdb7ddb8688635a.png)
(1)求椭圆的标准方程;
(2)求直线l的方程.
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2011·河北衡水·一模
8 . 设椭圆C1:
的左、右焦点分别是F1、F2,下顶点为A,线段OA的中点为B(O为坐标原点),如图.若抛物线C2:
与y轴的交点为B,且经过F1,F2点.
(Ⅰ)求椭圆C1的方程;
(Ⅱ)设M(0,
),N为抛物线C2上的一动点,过点N作抛物线C2的切线交椭圆C1于P、Q两点,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40c184c17e35b98fe8094cebe9522e1c.png)
(Ⅰ)求椭圆C1的方程;
(Ⅱ)设M(0,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af6ddf31b7d9225a4239883af72d153b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84a9dabb53dc826019fc8b6ae6d940c5.png)
![](https://img.xkw.com/dksih/QBM/2015/6/3/1572117314371584/1572117320482816/STEM/4c97a86bc0244a068037e08e9f9d0ebf.png?resizew=287)
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9 . 给定椭圆
,称圆心在坐标原点
,半径为
的圆是椭圆
的“伴随圆”. 若椭圆C的一个焦点为
,其短轴上的一个端点到
距离为
.
(1)求椭圆
及其“伴随圆”的方程;
(2)若过点
的直线
与椭圆C只有一个公共点,且
截椭圆C的“伴随圆”所得的弦长为
,求
的值;
(3)过椭圆C“伴椭圆”上一动点Q作直线
,使得
与椭圆C都只有一个公共点,试判断直线
的斜率之积是否为定值,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da001dad7941e6c9858637d7b62cec59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/996dadd8544543637bd2d32a1f79579e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9008010ed70af4a6e0d6c90cda8a17ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)过椭圆C“伴椭圆”上一动点Q作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
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